On integer partitions corresponding to numerical semigroups

Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions

Aggressive mammary carcinoma progression in Nrf2 knockout mice treated with 7,12-dimethylbenz[a]anthracene

<p>Abstract</p> <p>Background</p> <p>Activation of nuclear factor erythroid 2-related factor (Nrf2), which belongs to the basic leucine zipper transcription factor family, is a strategy for cancer chemopreventive phytochemicals. It is an important regulator of genes induced by oxidative stress, such as glutathione S-transferases, heme oxygenase-1 and peroxiredoxin 1, by activating the antioxidant response element (ARE). We <it>hypothesized </it>that (1) the citrus coumarin auraptene may suppress premalignant mammary lesions via activation of Nrf2/ARE, and (2) that Nrf2 knockout (KO) mice would be more susceptible to mammary carcinogenesis.</p> <p>Methods</p> <p>Premalignant lesions and mammary carcinomas were induced by medroxyprogesterone acetate and 7,12-dimethylbenz[a]anthracene treatment. The 10-week pre-malignant study was performed in which 8 groups of 10 each female wild-type (WT) and KO mice were fed either control diet or diets containing auraptene (500 ppm). A carcinogenesis study was also conducted in KO vs. WT mice (n = 30-34). Comparisons between groups were evaluated using ANOVA and Kaplan-Meier Survival statistics, and the Mann-Whitney U-test.</p> <p>Results</p> <p>All mice treated with carcinogen exhibited premalignant lesions but there were no differences by genotype or diet. In the KO mice, there was a dramatic increase in mammary carcinoma growth rate, size, and weight. Although there was no difference in overall survival, the KO mice had significantly lower mammary tumor-free survival. Also, in the KO mammary carcinomas, the active forms of NF-κB and β-catenin were increased ~2-fold whereas no differences in oxidized proteins were observed. Many other tumors were observed, including lymphomas. Interestingly, the incidences of lung adenomas in the KO mice were significantly higher than in the WT mice.</p> <p>Conclusions</p> <p>We report, for the first time, that there was no apparent difference in the formation of premalignant lesions, but rather, the KO mice exhibited rapid, aggressive mammary carcinoma progression.</p

Mock theta functions and related combinatorics

In this paper we add to the literature on the combinatorial nature of the mock theta functions, a collection of curious $q$-hypergeometric series introduced by Ramanujan in his last letter to Hardy in 1920, which we now know to be important examples of mock modular forms. Our work is inspired by Beck's conjecture, now a theorem of Andrews, related to Euler's identity: the excess in the number of parts in all partitions of $n$ into odd parts over the number of partitions of $n$ into distinct parts is equal to the number of partitions with only one (possibly repeated) even part and all other parts odd. We establish Beck-type identities associated to partition identities due to Andrews, Dixit, and Yee for the third order mock theta functions $\omega(q), \nu(q)$, and $\phi(q)$. Our proofs are both analytic and combinatorial in nature, and involve mock theta generating functions and combinatorial bijections

The heterologous expression of a <i>Glycine max</i> homolog of NONEXPRESSOR OF PR1 (NPR1) and α-hydroxynitrile glucosidase suppresses parasitism by the root pathogen <i>Meloidogyne incognita</i> in <i>Gossypium hirsutum</i>

<p>Experiments in <i>Glycine max</i> (soybean) identified the expression of the salicylic acid signaling and defense gene NONEXPRESSOR OF PR1 (NPR1) in root cells (i.e., syncytium) parasitized by the plant parasitic nematode <i>Heterodera glycines</i> undergoing the process of resistance. Gm-NPR1-2 overexpression in <i>G. max</i> effectively suppresses parasitism by <i>H. glycines</i>. The heterologous expression of Gm-NPR1-2 in <i>Gossypium hirsutum</i> impairs the ability of the parasitic nematode <i>Meloidogyne incognita</i> to form root galls, egg sacs, eggs and second-stage juvenile (J2) nematodes. In related experiments, a <i>G. max</i> β-glycosidase (Gm-βg-4) related to <i>Lotus japonicus</i> secreted defense gene α-hydroxynitrile glucosidase LjBGD7 suppresses <i>M. incognita</i> parasitism. The results identify a cumulative negative effect that the transgenes have on <i>M. incognita</i> parasitism and demonstrate that the <i>G. max</i>–<i>H. glycines</i> pathosystem is a useful tool to identify defense genes that function in other agriculturally relevant plant species to plant parasitic nematodes with different strategies of parasitism.</p